What's the best way to approach a sincere Christian relative about reality without starting an argument?

Atheist View

Suggest a book exchange: you will read any one book they send you, if they will read any one book you send them.

When arranging this, tell them if they want to talk about either book (if they want to know what you thought of the one they sent you, or they want to ask you questions about the one you sent them), they only have to ask you (and that they can ask you anything else they like as long as they are polite about it), but that you won't bring it up again if they don't.

Just be sure you communicate to them from the start that you cherish facts and logic, so in any discussions that ensue, you will always demand to see their evidence and sources (and you'll check them), and you will always pick at any faulty logic you see in what they say, so they should be prepared for that, and respectful of the fact that this is your religious belief: to follow sound evidence and logic, and question bad evidence and logic.

All this assumes they already know you are an unbeliever. Otherwise, you must politely come out first, making it known that you have strong moral beliefs that compel you to follow where the evidence leads, and that the evidence has led you to unbelief, but you still believe in kindness and honesty. Only if relations then remain cordial would proposing a book exchange be a good idea, and only after a sufficient period of adjustment (never right away).

Often, they will flatly turn down the book exchange offer. Many people are afraid to discover they are wrong and thus refuse even to look at the evidence. Leave them alone. There is nothing you can do for them, other than be a role model for love, honesty, and courage without god.